Investment hysteresis under stochastic interest rates

Jose Carlos Dias and Mark B. Shackleton∗

4th February 2005

Abstract

Most decision making research in real options focuses on revenue uncertainty assuming

discount rates remain constant. However for many decisions, revenue or cost streams are

relatively static and investment is driven by interest rate uncertainty, for example the

decision to invest in durable machinery and equipment. Using interest rate models from

Cox et al. (1985b), we generalize the work of Ingersoll and Ross (1992) in two ways.

Firstly we include real options on perpetuities (in addition to ”zero coupon” cash flows).

Secondly we incorporate abandonment or disinvestment as well as investment options and

thus model interest rate hysteresis (parallel to revenue uncertainty, Dixit (1989a)). Under

stochastic interest rates, economic hysteresis is found to be significant, even for small sunk

costs.

Key Words: real options, interest rate uncertainty, perpetuities, investment hysteresis.

JEL: G31; D92; D81; C61.

∗Dias is with ISCTE and ISCAC, Portugal. Shackleton is with Department of Accounting and

Finance at Lancaster University, LA1 4YX, United Kingdom. Tel: +44 (0) 1524 594131, e-mail:

m.shackleton@lancaster.ac.uk. Dias gratefully acknowledges the financial support of the PRODEP III Pro-

gramme. Corresponding author: Jose Carlos Dias, Department of Finance, Instituto Superior de Contabilidade

e Administracao de Coimbra, Quinta Agrıcola, Bencanta, 3040-316 Coimbra, Portugal. Tel: +351 919337791,

e-mail: jdias@iscac.pt.

1

1 Introduction

In the seminal book of Dixit and Pindyck (1994) it is argued that most major investments

share three important characteristics in varying degrees: (1) the investment is partially or

completely irreversible. By definition, an investment entails some sunk cost since it cannot be

totally recouped if the action is reversed later; (2) there is uncertainty about the future rewards

from the investment. Investment decisions are made in an economic environment with ongoing

uncertainty, where information arrives gradually; and (3) firms have some leeway about the

timing of investment. Usually, an investment opportunity does not disappear if it is not taken

immediately. As a result, firms have to decide whether to invest as well as when to invest. It is

the interaction between these characteristics that will determine the firm’s optimal investment

decision. Moreover, when these conditions are met, there is some positive value of waiting for

better information. Under these premises, an investment expenditure involves exercising the

option to invest, i.e., an option to optimally invest at any time in the future. As a result, the

full cost of investment must be the sum of two terms: the cost of investment itself and the

opportunity cost value of the lost option.

Investment rules that ignore these characteristics can lead to very wrong answers1. For

example, McDonald and Siegel (1986) show that the value of this opportunity cost can be

large, even for moderate levels of uncertainty. The importance of such characteristics was

also made clear by Ross (1995) in his description on the good, the bad, and the ugly of the

traditional net present value rule (NPV)2. For example, one of the implicit assumptions in the

traditional NPV rule is that either the investment decision is reversible, and thus firms can

undo some of their investment decisions and recover at least a fraction of the investment costs,

or the investment decision is irreversible, in the sense that it is a now or never decision. This

1Ignoring these characteristics means that the specific stream of cash flows are completely described in a

certain context, that is, the investment projects are treated simply as a stream of cash flows that are exogenously

determined without any implicit or explicit contingency admitted. Therefore, managers do not have the ability

to affect the cash flows of the project over time. In fact, they do not have anything to manage because flexibility

is not taken into account.2The good, the bad, and the ugly of the NPV implies, respectively, rejecting an investment when it should

be rejected, rejecting an investment when it should be accepted and accepting an investment when it should be

rejected.

2

means that if the firm does not invest now is not able to do it in the future. Although some

investments can be reversible, the majority of them are at least partly irreversible because

firms cannot recover all the investment costs. On the contrary, in some cases additional costs

of detaching and moving machinery may exist. Since most of the capital expenditures are firm

or industry specific they cannot be used in a different firm or different industry. Therefore, they

should be considered as largely a sunk cost. But, even if the capital expenditures would not be

firm or industry specific, they could not be totally recovered due to the ”lemons” problem of

Akerlof (1970). Hence, major investment costs are in a large part irreversible.

The irreversible assumption was first introduced by Arrow (1968) and Nickell (1974a,b),

among others, but under a certainty framework. Several irreversible investment decision prob-

lems with various types of uncertainty were considered afterwards.

Section 2 of this paper describes the uncertain interest rate processes used while Section 3

details the behaviour near the natural zero interest rate boundary. Section 4 develops analytical

solutions for the perpetuity as far as possible while Section 5 presents numerical solutions.

Section 6 solves the hysteresis problem numerically for the same process as Ingersoll and Ross

(1992). Section 7 concludes.

2 Term structure dynamics for spot rates with mean re-

version

The first models attempting to describe the term structure dynamics for spot rates did not

restrict spot rates to non-negative values, that is, they assumed a positive probability of negative

interest rates. For example, Merton (1973a) models the spot rate as an arithmetic Brownian

motion and Vasicek (1977) models it as an Ornstein-Uhlenbeck process. Since these processes

generate normally distributed interest rates, these can become negative with strictly positive

probability. This may make it applicable to real interest rates, but less appropriate for nominal

interest rates. Therefore, the applicability of these models to describe spot rate dynamics is

limited. Rendleman and Bartter (1980) assume that the instantaneous riskless interest rate

follows a geometric Brownian motion3. Therefore, they assume that the short-term behavior of

3Dothan (1978) also assume that the rate of interest follows a geometric Brownian motion, but with no drift.

3

interest rates is like the behavior of stock prices under the Black and Scholes (1973) framework.

Although this diffusion process precludes negative interest rates (similar to the limited liability

effect of stock prices), there is one important difference between interest rates and stock prices.

Thus, empirical evidence on interest rate behaviour seems to indicate that interest rates are

pulled back to some long-run mean value over time, a phenomenon that is usually called mean

reversion4. Apart from empirical evidence, this behavior also has economic intuition. For

example, Hull (2002, chap. 23) argues that when interest rates are high, the economy tends

to slow down and borrowers will require less capital (e.g., mean reversion tends to imply a

negative drift) and, as a result, interest rates decline. When interest rates are low, borrowers

tend to require more funds (e.g., mean reversion tends to imply a positive drift) and, therefore,

the interest rates tend to rise as a result of a high capital demand. Therefore, other models

were developed to describe the term structure dynamics for spot rates with mean reversion5.

The first models to assume that the instantaneous riskless interest rate follows a diffusion

process with a reverting mean were Vasicek (1977) and Cox et al. (1985b)6. The classical setting

in which the term structure dynamics for spot rates follows a mean-reverting diffusion process

can be stated as follows:

drt = κ(θ − rt)dt + σrtγdWt , r(0) = r0 (1)

where κ, θ, σ and γ are constants, dWt is a standard Gauss-Wiener process and it is usually

assumed that γ = 0 or 1/2 ≤ γ ≤ 1 (see, for example, Geman and Yor (1993)). From this

general setting, several models can be analyzed. Chan et al. (1992) presents one of the first

attempts to do an empirical comparison of alternative models of the short-term interest rate.

4For a more detailed description of the empirical evidence on this issue, see, for example, Campbell et al.

(1997, chap. 11) and the references contained therein.5It should be noted that our focus is on the so-called single-factor time-homogeneous models class. This

name comes from the fact that in a single-factor equilibrium model, the process for the interest rate involves

only one source of uncertainty. Thus, the instantaneous drift and the instantaneous volatility are assumed to be

functions of the interest rate, but are independent of time. Different assumptions taken for the instantaneous

drift and volatility will, of course, lead to different models. For a more detailed description of this type of

models as well as other interest rate models see, for example, Bjork (2004) and Musiela and Rutkowski (1998).6It should be noted that although this later paper was published only in 1985, it was already circulating in

the academia since 1976 as a working paper.

4

One of the key results found in their study is that the value of γ is the most important feature

differentiating dynamic models of the short-term riskless rate. Moreover, they showed that

models which allow γ ≥ 1 can capture the dynamics of the short-term interest rate better

than those with γ < 1. This is because the volatility of the process in the short-run is highly

sensitive to the level of the interest rate. Under these premises, they argue that the model used

by Cox et al. (1980) to study variable-rate securities, where γ = 3/2 and the process has no

drift, or the model of Dothan (1978) perform better relative to other more well-known models.

Of course the lack of mean reversion simplifies term structure models and there is an apparent

weak evidence of mean reversion in the short-term rate as it is shown in Chan et al. (1992). In

the long-run, however, the empirical evidence on interest rate behaviour seems to indicate that

there is a mean reversion effect, (as stated before). Therefore, for optimal investment decisions

or capital budgeting problems in a competitive environment under interest rate uncertainty

the assumption of no mean reversion of interest rates may not be adequate. Moreover, the

conclusion of Chan et al. (1992) that the models of Dothan (1978) and Cox et al. (1980)

perform better than the Vasicek or the standard Cox-Ingersoll-Ross square-Gaussian models

must be interpreted with caution, since the diffusions of the Dothan and the Cox-Ingersoll-Ross

variable-rate models are not ergodic (see Rogers (1995)).

Taking these issues into account, the cases where γ = 0 and γ = 1/2 are of primary interest

because they lead to analytic solutions, at least for finite maturities7. Both cases will result in

single-factor models and may be criticized on these grounds. By passing to multi-factor models

one should get an improved fit to observed prices, but there is a heavy price to pay since the

resulting partial differential equation would have a higher dimension. If our objective were to

calculate prices of some interest rate derivatives then other factors could be included in the

analysis in order to match observed prices. However, since our focus is on the effects of interest

7As it was stated before, one drawback of assuming γ = 0 is that the short-term interest rate can, in

some circumstances, become negative which is not a very appealing feature. The case where γ = 1 is used by

Brennan and Schwartz (1980) to analyze convertible bonds and Courtadon (1982) to price options on default-

free bonds. The main advantage of this mean-reverting process (with γ = 1) is that it presents a boundary at

zero. Therefore, it avoids the negative interest rates problem. However, analytical solutions can be found only

in very special cases and, thus, numerical methods have to be used since all the analytical tractability is lost.

Moreover, as it is suggested by Rogers (1995) none of these two models, that also possess the mean reversion

feature, appears to be conclusively superior to the Vasicek (1977) and the Cox et al. (1985b) models.

5

rate uncertainty on investment decisions a single-factor model of interest rates seems adequate.

2.1 The Vasicek term structure model

If we set γ = 0 we obtain the one-dimensional Ornstein-Uhlenbeck process that become familiar

in finance due to Vasicek (1977). Thus, its short-term structure dynamics for the interest rates

is a particular form of equation (1) that we will state as follows:

drt = κ(θ − rt)dt + σdWt , r(0) = r0 (2)

where κ is the parameter that determines the speed of adjustment (reversion rate), i.e., it

measures the intensity with which the interest rate is drawn back towards its long-run mean,

θ is the long-run mean of the instantaneous interest rate (asymptotic interest rate), σ is the

volatility of the process, rt is the instantaneous interest rate and dWt is a standard Gauss-

Wiener process. Moreover, it is usually assumed that κ, θ and σ are strictly positive constants.

The drift term of the process, κ(θ − rt), is a restoring force which always pull the stochastic

interest rate toward a long-term value (or mean value) of θ. The diffusion term of the process,

σ2, represents the variance of instantaneous changes in the interest rates.

Under this framework, the fundamental partial differential equation to price a default-free

discount bond, P , promising to pay one unit of capital at time T , is equal to:

1

2σ2∂2P (r)

∂r2+ κ(θ − r)

∂P (r)

∂r+

∂P (r)

∂T− λσ

∂P (r)

∂r− rP (r) = 0 (3)

where λ represents the market price of risk in the Vasicek’s model. This partial differential

equation has to be solved subject to boundary condition P (r, T, T ) = 1. Under this terminal

condition, the solution for the bond prices is given as8:

P (r, t, T ) = eA(t,T )−B(t,T ) r(t) (4)

where

8Using Vasicek’s interest rate model and bond pricing formulas, Jamshidian (1989) provide analytical solu-

tions for both European options on pure discount bonds and European options on coupon-bearing bonds. The

respective European put options can be obtained using the put-call parity relationship. Since these issues are

beyond the scope of this paper we will not reproduce the results here.

6

B(t, T ) =1− e−κ(T−t)

κ(5a)

A(t, T ) =(B(t, T )− T + t)(κ2θ − σ2/2)

κ2− σ2B(t, T )2

4κ(5b)

This model is one of the very popular models for the term structure dynamics of interest

rates, because it allows many of the bond and derivative prices to be computed easily in closed

form. The Gaussian feature of the Vasicek’s model has, however, the drawback that the interest

rate process occasionally take negative values. This undesirable feature may imply that bond

prices can grow exponentially which is absurd (see, for example, Rogers (1995)). As a result,

the prices of long-term pure discount bonds can exceed their face value, which is obviously a

spurious result (see Rogers (1996)). Therefore, we will use the Cox-Ingersoll-Ross framework to

price perpetuities, although the use of the Vasicek’s framework would lead to a mathematically

simple valuation of perpetuities.

2.2 The Cox-Ingersoll-Ross (CIR) term structure model

The well-known valuation framework of asset pricing in a continuous-time competitive economy

developed by Cox et al. (1985a) has been the basis for many equilibrium models of contingent

claims valuation. For example, the general equilibrium approach to term structure modelling

developed by Cox et al. (1985b) is an application of their more general equilibrium framework.

In their single factor model of the term structure of interest rates they assume that the interest

rate dynamics can be expressed as a diffusion process known as the mean-reverting square root

process:

drt = κ(θ − rt)dt + σ√

rtdWt , r(0) = r0 (6)

where the parameters of the model have the same meaning as before. The drift term of the

process, κ(θ − rt), is again a restoring force which always pulls the stochastic interest rate

toward a long-term value of θ. The diffusion term of the process, σ2rt, represents the variance

of instantaneous changes in the interest rates.

Due to the presence of the square-root in the diffusion coefficient, this process takes only

positive values. It can reach zero if σ2 > 2κθ, but it never becomes negative. If 2κθ ≥ σ2,

7

the upward drift will dominate preventing rt of reaching the origin9. Thus, we can sum up

the properties of this stochastic process as follows: (i) if the interest rate starts positive it

can never subsequently become negative; (2) if the interest rate hits zero, it can subsequently

become positive; (3) the absolute variance of the interest rate increases as the interest rate itself

increases; (4) there is a steady state distribution for the interest rate.

The distribution of the future interest rates has some properties, with respect to expected

value and variance, that are worthwhile to review10. If κ approaches infinity, the long-run mean

goes to θ and the variance to zero. But, as κ approaches zero the mean value goes to the current

interest rate and the variance to σ2r(t)× (s− t), where t and s denotes current time and future

time respectively. Thus, taking the limiting case of κ approaching zero, we are setting a new

process without drift, i.e., with a zero-expected change in the interest rates. However, as it was

pointed out by Ingersoll and Ross (1992), the qualitative properties of the zero-drift process

would also be true with their more general drift term.

Under this framework, the fundamental partial differential equation to price a default-free

discount bond, P , promising to pay one unit of capital at time T , is equal to:

1

2σ2r

∂2P (r)

∂r2+ κ(θ − r)

∂P (r)

∂r+

∂P (r)

∂T− λr

∂P (r)

∂r− rP (r) = 0 (7)

with the boundary condition P (r, T, T ) = 1. Since the first three terms of equation (7), which

come from Ito’s formula, represent the expected price change for the bond, the expected return

on the bond is r + (λr ∂P (r)∂r

× 1P). The factor λr represents the covariance of changes in the

interest rate with percentage changes in optimally invested wealth and λ is the market risk

parameter or price of interest rate risk. Due to the fact that ∂P (r)∂r

< 0, positive premiums will

exist if λ < 0, i.e., if this covariance is negative. The discount bond price is then equal to11:

P (r, t, T ) = A(t, T ) e−B(t,T ) r(t) (8)

9Additional details concerning this relationship will be discussed in the next section.10See Cox et al. (1985b) for additional details.11Cox et al. (1985b) further extended their term structure model to price European call and put options

written on discount bonds. Their formulae is also suitable to price American call options on discount bonds

because they will never be exercised before maturity, given the absence of interim coupons on these bonds (see

Merton (1973a)).

8

where

A(t, T ) =

[2ωe[(κ+λ+ω)(T−t)]/2

(ω + κ + λ)(eω(T−t) − 1) + 2ω

]2κθ/σ2

(9a)

B(t, T ) =2(eω(T−t) − 1)

(ω + κ + λ)(eω(T−t) − 1) + 2ω(9b)

ω =[(κ + λ)2 + 2σ2

]1/2(9c)

This model is also one of the very popular models for the term structure dynamics of interest

rates. However, the criticism that was applied to Vasicek’s arbitrage model does not apply to the

Cox et al. (1985b) intertemporal general equilibrium term structure model, because the latter

does not allow negative interest rates which is a desirable and more realistic feature for the term

structure dynamics of interest rates (see Rogers (1995)). Also, it appears that a square-root

process is more suited that an Ornstein-Uhlenbeck process, because there is empirical evidence

(e.g., Chan et al. (1992)) showing that the proportionality factor (the γ parameter of equation

(1)) assumed in the CIR model might be even to weak, since it is of order 1.5 on the US T-Bill

market. Therefore, the valuation of perpetuities under the CIR framework seems more suited

than under the Vasicek’s model.

3 Boundary conditions on CIR’s interest rate dynamics

The so-called Bessel processes play a key role in financial mathematics because of their strong

relation with several diffusion processes that are usually used in finance. Examples of these

diffusions are the geometric Brownian motion, the Ornstein-Uhlenbeck process and the square-

root process. Although this later diffusion was firstly introduced in finance by Cox et al.

(1985b), several of its properties were already studied before as squared Bessel processes. These

processes are, by definition, Markov processes12. One of the key issues of the square-root

diffusion is the role played by the term κθ, which is closely related with the dimension δ of

a squared Bessel process (δ = 4κθ/σ2), and have important implications for the boundary

conditions of the problem (see, for example, Feller (1951); for a complete description of the

12This means that the probability distribution for all future values of the process depends only on its current

value and is not affected by past values of the process or by any other current information.

9

boundary classification for one-dimensional diffusions see Karlin and Taylor (1981, chap. 15)).

The values of the function both at r = 0 and r = +∞ are of particular interest when we

are dealing with interest rate problems. From these two points, only the first one deserves

particular attention since no key phenomenon occurs at infinity, because the infinite point is

a natural boundary for all specifications of κθ. But, at r = 0 the specification of the κθ term

completely changes the behaviour of the problem.

Three important properties are of particular interest13: (i) if 2κθ ≥ σ2, r = 0 is an entrance,

but not exit, boundary point for the process. This means that 0 acts both as absorbing and

reflecting barrier such that no homogeneous boundary conditions can be imposed there. Thus,

the origin is inaccessible and the CIR process stays strictly positive14; (ii) if 0 < 2κθ < σ2, r = 0

is a reflecting boundary (exit and entrance), i.e., 0 is chosen to be an instantaneously reflecting

regular boundary; (iii) if κθ = 0, r = 0 is a trap or an absorbing point and no boundary

condition can be imposed there. Thus, when the CIR diffusion process hits 0 it is extinct, i.e.,

it remains at 0 forever (absorbing or exit boundary). This last property is implausible for a spot

rate process because it predicts that the interest rate will remain forever at 0 once this level

has been reached. Thus, a reflecting boundary model seems quite reasonable for an interest

rate process, but an absorbing boundary model does not15.

13We are assuming that κθ ≥ 0, σ > 0 and the CIR diffusion process is valued in [0, +∞). Square-root

diffusions with these characteristics have been well studied (see, for example, Revuz and Yor (1999, chap. 11)).

But it seems also natural to consider the case where the dimension of the process is negative, resulting in

κθ < 0, or even extend it to negative starting points (see, for example, Going-Jaeschke and Yor (2003)). This

last extention may be useful for some applications, but for the interest rate dynamics is of no interest.14This means that an entrance boundary cannot be reached from the interior of the state space that is

considered, [0, +∞) in this case. Thus, if the process starts with a positive interest rate, it cannot subsequently

become negative or reach the origin. However, it is possible for some other applications to assume that the

state variable begins at the left boundary, but then the stochastic process will move to the interior of the state

space and will not return to the entrance boundary again.15For example, a slowly reflecting boundary, in which the interest rate can remain at 0 for a finite period of

time before returning to positive values again, is more plausible than an absorbing boundary at 0.

10

4 Valuation of perpetuities under CIR stochastic inter-

est rates

Following Cox et al. (1985a,b), the price of any interest-rate contingent claims satisfies the

following partial differential equation:

1

2σ2r

∂2F (r)

∂r2+ κ(θ − r)

∂F (r)

∂r+

∂F (r)

∂T− λr

∂F (r)

∂r− rF (r) + C(r, t) = 0 (10)

This equation is similar to equation (7). The only difference is the new term C(r, t) which

represents the cash rate paid out to the claim16. For the valuation of a default-free discount

bond C(r, t) = 0, but for a perpetuity its value is 1 since a perpetuity is a default-free financial

instrument that pays a constant stream of one unit of capital p.a.17. In addition, for a perpetuity

the term ∂F (r)∂T

will vanish as T goes to infinity. Thus, equation (10) can be restated as:

1

2σ2r

∂2F (r)

∂r2+ κ(θ − r)

∂F (r)

∂r− λr

∂F (r)

∂r− rF (r) + 1 = 0 (11)

As it was already stated, equation (8) is the solution to the partial differential equation

(7), which allow us pricing a default-free discount bond, i.e., pricing a bond for finite maturity.

Although this solution was first introduced in finance by Cox et al. (1985b), the formula was

already obtained by Pitman and Yor (1982) but in a different context (see, for example, Delbaen

(1993) and Geman and Yor (1993)). Thus, the price at time t = 0 of a zero coupon bond

maturing at time T is also equal to:

P (r, 0, T ) = EQ0[e−

∫ T0 r(s) ds

]= A(0, T ) e−B(0,T ) r(0) (12)

where EQ0 denotes the expectation under the risk-neutral probability Q (or martingale measure

Q), at time t = 0, with respect to the risk-adjusted process for the instantaneous interest rate

that can be written as the following stochastic differential equation:

drt = [κθ − (λ + κ) rt] dt + σ√

rt dWt (13)

16We also changed the function notation to distinguish the value of a default-free discount bond, P (r), from

the value of a perpetuity, F (r).17Another common name for a perpetuity is consol.

11

and where dWt is a standard Brownian motion under Q. It should be noted that option

pricing analysis usually resort in the so-called risk-neutral valuation which is essentially based

in replication and continuous trading arguments18. However, the interest rate r is not the price

of a traded asset, since there is no asset on the market whose price process is given by r. This

means that the present framework is somewhat more complicated than a Black-Scholes setting

due to the appearance of the market price of risk λ, which is not determined within the model.

We see that the value at time t = 0 of a zero coupon bond with maturity date T is given as

the expected value of the final payoff of one dollar discounted to present value. This expected

value is stated by equation (12), but in this case the expectation is not to be taken using the

objective probability measure P . Instead, a martingale measure Q must be used to denote

that the expectation is taken with respect to a risk-adjusted process, where the risk adjustment

is determined by reducing the drift of the underlying variable by a factor risk premium λr.

Therefore, the risk-adjusted drift of the interest rate square-root process is denoted by the

term [κθ − (λ + κ) rt]. It should also be emphasized that although risk premiums for interest

rates may be introduced, they cannot be observed or measured. Thus, in order to be able to

solve such problem, an exogenously given λ must be specified. Moreover, the risk factor term

to be introduced is determined by things such as the forms of risk aversion possessed by the

various agents on the market. This means that if one makes an ad hoc choice of λ = 0, then

he is implicitly making an assumption concerning the aggregate risk aversion on the market19.

Under this framework, we can set the value of a perpetuity, that we will denote as F (r), as

follows20:

18For example, it is possible to compute the Black-Scholes arbitrage free prices using such arguments and a

risk-neutral valuation approach, because there is a risk-neutral probability measure Q equivalent to the real

world probability measure P [see, for example, Cox and Ross (1976), Harrison and Kreps (1979) and Harrison

and Pliska (1981)]. Since in the case of most real option problems the underlying asset is often not traded, for

contingent-claims valuation it is not needed to invoke replication and continuous trading arguments. Instead,

it is usually assumed that an intertemporal capital asset pricing model holds, like the one of Merton (1973b),

and then a general equilibrium model, such as the one of Cox et al. (1985a), is developed.19For a detailed technical exposition regarding these issues see, for example, Bjork (2004).20The valuation of perpetuities using the methodology of Bessel processes under stochastic interest rates

within the CIR’s framework can be found in Delbaen (1993), Geman and Yor (1993) and Yor (1993). Several

mathematical properties are also investigated.

12

F (r) = EQ0[ ∫ ∞

0

e−∫ t0 r(s) ds dt

]=

∫ ∞

0

P (r, 0, t) dt (14)

As we will see later, we need to use the first derivative of the perpetuity function. Differentiation

under the integral sign is allowed, even when a limit is infinite, and this gives us:

F ′(r) =d

dr

∫ ∞

0

P (r, 0, t) dt =

∫ ∞

0

∂P (r, 0, t)

∂rdt = −

∫ ∞

0

A(0, t)B(0, t) e−B(0,t) r(0) dt (15)

In order to compute the value of a perpetuity and its derivative we will use the parameter values

taken from the empirical work of Chan et al. (1992). Their values will be considered as our

base case parameter values. Additionally, we are also interested in the special case where the

term κθ = 0. As it was pointed out by Ingersoll and Ross (1992) there is no consensus about

the appropriate λ value to use. As a result, throughout this work we will generally consider

no term premia (λ = 0), yet in some specific cases we will use some positive term premia for

comparative reasons only. The parameter values for both cases are presented in Table 121.

Table 1: Parameter values for the base and special cases.

Parameter Base Case Value Special Case Value

κ 0.2339 0

θ 0.0808 0

σ 0.0854 0.0854

λ 0 0

21For the special case we are considering that both κ and θ are zero, but it is a sufficient condition that only

κ be zero to generate such case, since it is this parameter that plays a key role on the distribution of the future

interest rates. As Cox et al. (1985b) have shown as κ → 0 the conditional mean goes to the current interest rate

and the conditional variance of r(s) given r(t) (where s > t) goes to σ2 r(t)× (s− t). Therefore, this lead to the

single-factor pure diffusion process of Ingersoll and Ross (1992) that we will use as our special case. But θ also

plays a significant role even if the κ parameter is not zero. In this case, if we consider that θ = 0 it still would

not be possible to impose a boundary condition at r = 0. However, we would continue to have a mean-reverting

process but now with an asymptotic interest rate equal to zero.

13

5 Numerical integration of stochastic interest rates

Now it is necessary to present some numerical computations in order to understand the be-

haviour of the functions and the impact of considering, or not, a κθ term equal to zero and

different volatility and risk premium levels. Table 2 shows the values of both functions for

the base case parameter values, considering different maturities22 and volatilities and with

r(0) = 023.

Table 2: Values of the perpetuity function and the first

derivative of the perpetuity function using the base case

parameter values for different levels of volatility.

r(0) = 0.00

Function T σ = 0.03 σ = 0.0854 σ = 0.3

F (r) 100 16.136 16.595 21.275

F ′(r) 100 -52.888 -52.877 -52.913

F (r) 500 16.141 16.604 21.275

F ′(r) 500 -52.913 -52.913 -52.913

F (r) 1000 16.141 16.604 21.275

F ′(r) 1000 -52.913 -52.913 -52.913

F (r) ∞ 16.141 16.604 21.275

F ′(r) ∞ -52.913 -52.913 -52.913

The results from Table 2 seems to indicate that the use of a fixed number T in the upper limit

22It should be noted that with this approach the resulting formulae does still hold even if the upper bound of

the integral in F (r) is a fixed number T . For example, since the stochastic nature of interest rates is particularly

relevant for actuarial purposes, considering a fixed number T in the upper limit of the integral may be a better

description of the finite nature of human life for such applications.23We choose this interest rate since the values of the functions at this point are of particular interest for the

one-factor model that we are using, but similar computations can be done for any positive interest rate wanted.

For the special case, however, the value of the perpetuity at r = 0 diverges for infinity since the term κθ = 0.

For this reason it is not necessary to reproduce the results here.

14

of the integral, instead of using infinity, will not generate any problem for the base case. We

also have tried other interest rate values and we reach the same conclusions. Thus, considering

T = 500 or T = 1000 seems quite reasonable for the analysis and it will simplify the numerical

computations if we use this approach. Even the use of T = 100 will not produce to much

differences. But instead of using equations (14) and (15) to compute the value of a perpetuity

and its derivative, we can use, as an alternative, the analytic functions proposed by Delbaen

(1993) and Geman and Yor (1993). Using such formulae we also achieve the same values that

we present in the table when T = ∞. Therefore, the choice between one of the two ways to

compute perpetuities under the CIR framework is at the decision of the user. Using a finite

maturity may be useful in some insurance applications. However, for other applications such as

real options, where it is usually considered that the problems under study are time-independent,

any of the alternative approaches are quite suitable. Yet, the use of T = 500 or T = 1000 in

the first approach would generate similar results.

Figure 1 presents the value of a perpetuity as a function of the interest rate using the base

case parameter values. It has finite value and slope at zero because rates being stochastic do

not remain there for long. The value of the function at r = 0 is 16.604 and F ′(0) = −52.913. It

should be noted that for the particular case where r = 0, F ′(0) = −1/κθ (see Delbaen (1993)).

This shows why the value of F ′(0) is the same for different volatility levels (see Table 2) and

even for different risk premiums (see Table 3 that we will present below), because at r = 0 it

only depends on κ and θ. This also clearly demonstrates why it is not possible to impose a

boundary condition at r = 0 when the term κθ = 0.

Figure 2 presents the value of a perpetuity as a function of the interest rate using the

base case parameter values for different levels of volatility. Thus, a perpetuity is an increasing

function of the volatility level. However, the way the volatility level affects the perpetuity value

is not exactly the same when we consider, or not, that the term κθ = 0.

In Figure 3 we present the value of a perpetuity as a function of the interest rate volatility

using the base case parameter values with r(0) = 0.06 and two levels of risk premium λ. In this

case, the perpetuity value is increasing and strictly convex. But for the special case we have a

different behaviour as it is shown in Figure 4. Thus, for lower levels of volatility an increase in

the interest rate volatility level does not produce a significant impact on the perpetuity value.

15

This issue is even more significant for positive risk premiums. But for higher levels of volatility

an increase in the volatility level generates a sharply increase on the perpetuity value. This

behaviour is justified based on the fact that the volatility level plays a key role in the special

case, when compared with the base case, since the term κθ is zero24. We also used other values

for r(0) and we reached similar conclusions for both cases.

Until now we are considering only positive volatility levels. It is also interesting to compare

both cases with the most basic perpetuity that we can use in finance, the case where we have

a zero volatility level (also no mean reversion or risk premium). In this case the perpetuity

function is just F (r) = 1/r. Figures 5 and 6 present the value of a perpetuity as a function of

the interest rate using the base case values and special case values, respectively. The particular

case with zero volatility is just the case where F (r) = 1/r. Thus, it is easy to see that volatility

plays a key role for the special case. In addition, we can conclude that for very low levels of

interest rate volatility and moderate interest rate levels the use of a perpetuity of the form

F (r) = 1/r is quite reasonable if we are considering that there is no mean reversion effect.

But if the true process is mean-reverting this is not true. For lower levels of interest rates the

perpetuity value with zero volatility will be overvalued and for higher interest rates it will be

undervalued. The conclusion that we can take from this is that even when we are considering

low levels of interest rate volatility, the use of a perpetuity function of the type F (r) = 1/r can

produce gross errors for the analysis if the true generating rate process is mean-reverting.

Table 3 presents the values of the perpetuities and its derivatives for the base case parameter

values, considering different maturities and risk premiums and with r(0) = 0. Thus, pricing

perpetuities in a general equilibrium framework under different degrees of risk aversion is pos-

sible, but an additional unobserved parameter value is required, the market price of risk which

we assume to be constant. The values of perpetuities under the special case will, obviously,

diverge and it is not necessary to reproduce the results here. Once again, the use of T = 500 or

T = 1000 is quite acceptable, but the use of T = 100 is also perfectly reasonable especially for

positive risk premiums. We also have tried other interest rate levels and we achieve the same

conclusions about this issue.

24It should be noted that in this case, equation (8) will be simplified to P (r, t, T ) = e−B(t,T ) r(t) since

A(t, T ) = 1, resulting in a higher role of the interest rate volatility for the stochastic process.

16

Table 3: Values of the perpetuity function and the first

derivative of the perpetuity function using the base case

parameter values for different levels of the risk premium.

r(0) = 0.00

Function T λ = 0.0 λ = −0.1 λ = −0.2

F (r) 100 16.595 12.954 10.255

F ′(r) 100 -52.877 -52.912 -52.913

F (r) 500 16.604 12.954 10.255

F ′(r) 500 -52.913 -52.913 -52.913

F (r) 1000 16.604 12.954 10.255

F ′(r) 1000 -52.913 -52.913 -52.913

F (r) ∞ 16.604 12.954 10.255

F ′(r) ∞ -52.913 -52.913 -52.913

Figure 7 presents the value of a perpetuity as a function of the interest rate using the base

case parameters for different values of risk premium, the higher the risk premium the lower the

price. A perpetuity is a decreasing function of the risk premium level. In this case, the pattern

the risk premium level affects the perpetuity value is similar for both the base and special cases

(strictly decreasing and strictly convex), but it is obviously more pronounced for the later as

Figure 8 shows.

It is also interesting to analyse the behaviour of the perpetuity function under different

values of κ and θ. The perpetuity is a decreasing convex function of the speed of adjustment

for lower levels of interest rates (i.e., interest rates < θ), but it is an increasing concave function

for higher levels of interest rates (i.e., interest rates > θ) as it is shown in Figures 9 and 10.

The interest rate level of r = 0.1241 corresponds to the intersection point between the cases

with κ0 and κ2 resulting in a perpetuity value of 11.287. Thus, the value of the κ parameter is

highly relevant since different levels of speed of adjustment will lead to an intersection between

any two of the κ levels. This is particularly relevant for the choice of using an interest rate

process with or without mean reversion.

17

Figure 11 shows that the value of a perpetuity is a decreasing function of the asymptotic

interest rate θ.

As Figure 12 highlights there is an interest rate level for which the consideration of mean

reversion or no mean reversion makes the perpetuity value the same. A simple numerical

computation shows that under the parameters that we are using the intersection point is where

r = 0.3046 resulting in a perpetuity value of 6.684. Thus, ignoring the mean reversion effect

in order to gain simplicity, when the true generating rate process is mean reverting and its

existence is strongly documented in the vast empirical evidence that we have previously cited,

may cause gross errors for applications in which computing perpetuity values are crucial. For

instance, for the case where the mean reversion effect best describe the reality of the economy,

the use of a κθ term equal to zero will lead to a perpetuity value overvalued for interest rates

lower than 0.3046 and a perpetuity value undervalued for interest rates higher than 0.3046. This

issue is highly relevant for investment and disinvestment decisions of firms where the interest

rate uncertainty is a key factor for the decision, since the upper interest rate threshold (trigger

point that will induce the firm to disinvest) and the lower interest rate threshold (trigger point

that will induce the firm to invest) will be surely influenced by this fact. These ideas will be

analysed in the following sections.

6 Investment hysteresis without mean reversion

6.1 Perpetual investment and disinvestment opportunities

To concentrate on the effects of interest rates on investment decisions we use a particular model

of real interest rates. To do so, we follow the single-factor pure diffusion process of Ingersoll and

Ross (1992) assuming that changes in the instantaneous interest rate, r, satisfy the following

process (with a zero mean reversion coefficient):

drt = σ√

rt dWt (16)

where σ is constant. This is equivalent to the interest rate dynamics of the risk-adjusted

stochastic process drt = −λrt dt + σ√

rt dWt for risk-neutral pricing in the case of a nonzero

term premium λ where it is assumed that λ is constant and λ < 0 corresponds to positive risk

18

premiums. The process followed here restricts the more general mean-reverting drift process of

Cox et al. (1985b), a general equilibrium model of the term structure of default-free securities,

and that is an application of their general equilibrium framework for asset pricing in continuous-

time (see Cox et al. (1985a)). Since we want to focus on the effects of interest rate uncertainty

on the investment and disinvestment decisions, the Ingersoll and Ross (1992) process with a

zero expected interest rate change will allow the simplification of our analysis. But in the next

section we will also consider the general mean-reverting square root process to analyze the

impact of stochastic interest rates under mean reversion on those decisions. Additionally, we

will be able to understand the implications of considering, or not, the mean reversion effect for

capital budgeting purposes under stochastic interest rates.

According to Cox et al. (1985a,b), the price of any interest-rate contingent claims satisfies

the following partial differential equation (for the case where mean reversion is not considered):

1

2σ2r

∂2F (r)

∂r2− λr

∂F (r)

∂r+

∂F (r)

∂T− rF (r) + C(r, t) = 0 (17)

where C is the net cash paid out to the claim and λ measures the price of interest-rate risk.

If λ < 0 it corresponds to a case of positive risk premium. Throughout the analysis we will

assume that λ is a constant.

Following the ideas that underlies most of the real options’ framework, we assume a very

long time to maturity options. This technique was firstly raised by Merton (1973a) to obtain

closed-form solutions for the perpetual calls and puts options. Using this technique the problem

stated in equation (17) becomes time independent since the term ∂F (r)∂T

will vanish as T becomes

very long(

∂F (r)∂T

→ 0). In addition, cash is paid out at a rate of a dollar per annum for ever.

For this perpetual case, equation (17) reduces to an ordinary differential equation of the form:

1

2σ2r

∂2F (r)

∂r2− λr

∂F (r)

∂r− rF (r) + 1 = 0 (18)

Looking at equation (18) it is easy to see that it does not have constant coefficients since

they are dependent on r. But with a single change we can turn the problem easier. Thus

dividing both sides of the equation by r and rearranging we get:

1

2σ2∂2F (r)

∂r2− λ

∂F (r)

∂r− F (r) = −1

r(19)

19

Now, equation (19) is a linear nonhomogeneous constant coefficient equation. The general

solution to this equation, F (r), is the sum of the complementary solution, y(r), and the par-

ticular solution, Y (r). The general solution can be interpreted using the following economic

intuition. Y (r) is interpreted as the expected present value payoff if the state variable r is

allowed to fluctuate without any regulation or barrier control, while F (r) is interpreted in the

same way, but now with the stochastic process being regulated by some form of control (we

will discuss this issue below). Therefore, y(r) must represent the additional value of control.

A possible and natural lower barrier for an interest rate process would be r = 0, but for this

single-factor pure diffusion process such control is not possible because the term κθ is equal

to zero and the slope infinite. As a result, we have to determine the barriers, as well as the

constants of the complementary solution, numerically since no closed-form solution is available.

In our case, the barrier controls will be determined by the decision to invest or disinvest, that

will lead to a lower trigger point or lower barrier point (r) and an upper threshold or upper

barrier point (r). Thus, the solution to y(r) is the sum of two terms, where one of them will be

interpreted as the perpetual investment opportunity and the other interpreted as the perpetual

disinvestment opportunity. Since we want to consider models of investment and disinvestment

we will add a new state variable to the decision problem, a discrete variable that will indi-

cate if the firm is active (1) or idle (0). When we consider combined entry and exit decisions

simultaneously the firm will have, in each state, a call option on the other. For example, if

an idle firm exercise its option to invest, it will get an operating profit plus a call option to

abandon. Similarly, if an active firm exercises its option to abandon it will return to the idle

state and get a new option to invest. In such case, the values of an idle firm and an active firm

are interlinked and must be determined simultaneously. Although the combined entry and exit

strategy is the most interesting one to analyze, we will also consider the isolated strategies of

invest and disinvest for comparisons purposes.

The value of an idle or not active firm, F0(r), is obtained by the solution of the complemen-

tary function of equation (19):

1

2σ2∂2F0(r)

∂r2− λ

∂F0(r)

∂r− F0(r) = 0 (20)

and the value of an active firm, F1(r), is the solution of the entire equation (19):

20

1

2σ2∂2F1(r)

∂r2− λ

∂F1(r)

∂r− F1(r) = −1

r(21)

Let us now proceed with the solution of the complementary functions together, since they are

similar linear homogeneous equations with constant coefficients. Trying a solution of the form

F (r) = emr, we find that F ′(r) = memr and F ′′(r) = m2emr. Substitution yields:

(1

2σ2m2 − λm− 1

)emr = 0 (22)

Hence F (r) = emr is a solution of Equation (19) when m is a root of

1

2σ2m2 − λm− 1 = 0 (23)

or

φ(m) = m2 − vm− w = 0 (24)

where we define v = 2λ/σ2 and w = 2/σ2. The convergence condition of equation (24) is

w > 1 − v. Then, it turns out that φ(0) = −w < 0 and φ(1) = 1 − v − w < 0. Since

φ′′(m) = 2 > 0 it means that the auxiliary equation has two roots, where one of them must be

greater than one (we will call it a) and the other one must be less than zero (we will call it b).

The discriminant of the characteristic equation is positive, ∆ = v2 +4w > 0, which means that

the respective solutions are real. Therefore, the two roots can be written out as:

a =+v +

√v2 + 4w

2> 1 (25a)

b =+v −√v2 + 4w

2< 0 (25b)

Thus, we can write the general solution of equation (20) as:

F0(r) = C1ear + C2e

br (26)

and the general solution of equation (21) as:

F1(r) = C3ear + C4e

br + Y (r) (27)

21

where C1, C2, C3 and C4 are constants to be determined from boundary conditions. A simple

economic intuition tells us that for very high interest rate levels idle firms are not induced to

invest. Therefore, the option of activating the firm should be nearly worthless for this level

rates. As a result, we need that the constant C1 = 0 (associated with the positive root a). This

means that the expected net present value of making an investment in the idle state is:

F0(r) = C2ebr (28)

Since an idle firm is not operating does not have any return from the project yet. Therefore,

equation (28) is just the option value of a perpetual investment opportunity, IO(r). Over the

range interval of interest rates (r, ∞), an idle firm will not exercise its option to invest. To

simplify our analysis, we will consider that once the investment commitment has been made,

the investment project return is identical to a perpetuity making a continuous payment of one

unit over time. Thus, no additional resources or expenditures apart from the initial investment

are required to maintain the rights over the project or to sustain the project after it has been

accepted. A similar assumption is also used by Ingersoll and Ross (1992), but in their case the

project returns are identical to a T-period zero-coupon bond with a real face value of one dollar

since they are considering finite maturities, whereas we are considering infinite maturities.

This assumption implies that operating profits never become negative in our project. Such

assumption is also used by, among others, McDonald and Siegel (1986), Pindyck (1988) and

Bertola (1998).

The value of an active firm is the sum of two components, the expected present value of

the profits and an option value of terminating the project. We know that for very low interest

rates an active firm will be induced to continue its operations and not disinvest. Since the

value of the abandonment option should go to zero as r becomes very low, we must set C4 = 0

(associated with the negative root b). Therefore, the value of a firm for the active state is:

F1(r) = C3ear + F (r) (29)

where F (r) is the particular solution Y (r) of the ordinary differential equation (21). It follows

that a particular solution to this equation is, as it was already stated before, the value of a

22

perpetuity making a continuous payment of one unit over time, i.e,∫∞

0P (r, 0, t) dt25. Since the

perpetuity value represents the expected present value that can be obtained from the project

if it is maintained active forever, the remaining part of equation (29) must be the value of

a perpetual option to disinvest optimally, i.e., DO(r) = C3ear. Over the interest rate range

(0, r) an active firm will continue its operations, holding its option to abandon alive. It should

be emphasized that as a simplifying setting we are considering that the investment option, as

well as the option to disinvest, are both perpetual since it is possible to delay both decisions

in an infinite time frame, which gives them some resemblance with the perpetual calls and

puts of Merton (1973a). Finally, it should be noted that for an option to invest, as well as an

option to disinvest, have an economically meaningful interpretation they must be non-negative.

Therefore, we must have, respectively, C2 ≥ 0 and C3 ≥ 0. All the numerical simulations that

we will present later have shown that this is the case.

6.2 Option to invest

Let us suppose that an idle firm has an option to invest in a particular investment project

where interest rate uncertainty is a key factor for the decision to invest, but where, for now, the

disinvestment opportunity is not considered. Thus, the firm has to decide whether to continue

being idle or to enter in the market. If interest rates drop to a low level, the firm may be

induced to change an option to invest paying an investment cost I by a perpetuity making a

continuous payment of one unit over time. This investment cost is considered a sunk capital

cost since it cannot be totally recouped if the firm should decide to quit at a later date. This

issue can be explained using the ”lemons” problem of Akerlof (1970). The investment strategy

can be stated as follows:

I + IO(r) → F (r)

The optimal investment policy is determined using one value matching condition and one

smooth pasting condition (also called high contact condition)26. This yields a system of two

25Therefore, we use equation (14) as F (r) for the special case but we have to impose a fixed number T in the

upper limit of the integral because the κθ parameter is zero.26Our real options problems are of American-type nature since they are time-independent and, therefore, can

23

non-linear equations in two variables (C2 and r):

I + C2ebr =

∫ ∞

0

A(t)e−B(t)r dt (30a)

bC2ebr =

∫ ∞

0

−B(t)A(t)e−B(t)r dt (30b)

The economic intuition of the value matching condition (30a) is that the gain in value from

exercising the option to invest (i.e. exercising the control) is exactly equal to the cost of doing

so. If this condition was not respected, then arbitrage profits would be possible. The optimality

condition of the problem does not arise from the value matching condition, but from the smooth

pasting condition. This condition (stated by equation (30b)) says that the first derivative of the

option value function must have the same value before and after the option has been exercised,

i.e., the marginal utility must take the same value before and after the action has been taken27.

be exercised at any time before maturity. Thus, they are optimal stopping problems. The optimality conditions

for such problems were introduced in the financial economics literature by Samuelson (1965), McKean (1965)

and Merton (1973a). For a general treatment of such conditions in a simpler setting see, for example, Dixit

(1991b, 1993) and Dumas (1991).27It should be noted that in this case the smooth pasting condition is obtained using the first derivative of

the value function, and it holds only for the optimum policy. This happens since the investment cost (as well

as the ”disinvestment cost”) is a lump-sum cost, i.e., it originates a discrete adjustment and the stochastic

control problem is regulated using the first derivative of the state variable function. This is what is usually

called by stochastic impulse control (see, for example, Constantinides and Richard (1978) and Harrison et al.

(1983)). Optimal impulse control has been applied for example by Brennan and Schwartz (1985) to model the

opening and closing of a mine and by Grossman and Laroque (1990) for an optimal consumption and portfolio

choice problem under the presence of fixed transaction costs. There are situations, however, where linear costs

or proportional costs of adjustment exist and for such cases the first-order smooth pasting condition holds

for any given barrier control, which requires a higher form of tangency. As a result, a second-order smooth

pasting condition (using the second derivative of the value function) replaces the first-order smooth pasting

condition at the optimal barrier leading to what Dumas (1991) called ”super contact” condition. This is what

is usually called by stochastic instantaneous control, where the order of differentiation moves up, i.e., the value

matching condition involves the first derivative of the value function and the smooth pasting condition involves

the second derivative (see, for example, Harrison and Taylor (1978), Harrison and Taksar (1983) and Harrison

(1985)). Examples of such application include the dynamic equilibrium model of Dumas (1992) for two separated

countries where it is possible to trade goods and/or capital abroad under proportional transfer costs and the

investor’s portfolio choice problem under proportional transaction costs of Dumas and Luciano (1991).

24

If this condition failed, then moving the critical interest rate would raise the value of the option.

The state variable of stochastic control problems referred in the previous footnote follows

a Brownian motion or a geometric Brownian motion. In our problem, the state variable r

follows a mean reverting square-root process in the base case and a zero-drift pure diffusion

process in the special case. However, as Dixit (1991b) points out the general framework of

optimal stochastic control can be adapted for other stochastic processes. Therefore, we will

have a stochastic impulse control problem for our both cases since the investment cost and the

disinvestment proceeds are discrete adjustments that are made at a discrete point in time, i.e.,

the moment where the decision is taken.

In order to obtain the values of C2 and r we can solve the system of the two non-linear

equations numerically, or, as an alternative, we can transform the system of two equations in

just one equation inverting the matrix system. Then, the lower threshold can be computed

solving the following equation:

I = F (r)− F ′(r)b

(31)

Although we can reduce the system to just one equation we still have a non-linear equation

and, as a result, we have to use numerical simulations to obtain the lower threshold. Once

we have the value of the lower trigger point it is extremely easy to compute the value of the

corresponding constant. For an idle firm, no action is taken until r reaches a lower barrier r,

when it is optimal to switch the value of the option to invest plus an investment cost by a

perpetuity value. The corresponding numerical simulations will be presented below.

6.3 Option to disinvest

Let us now suppose that an active firm is operating and its payoff is a perpetuity making a

continuous payment of one unit over time. But if interest rates start rising to very high rates

the firm may be induced to temporarily shut down or even abandon the project. If a project

is closed temporarily it turns out that the firm will incur some fixed maintenance costs, but

may be opened up again without having to pay again entry costs, i.e., I28. If the project is to

28It should be noted that if the maintenance costs are zero then it is never optimal to abandon a closed project

as long as there is a real possibility that it will be optimal to reopen the project again. But this assumption of

25

be permanently abandoned it will incur no maintenance costs, but if the firm wants to enter

again in the market has to pay a new lump-sum cost I. This possibility (i.e., a re–entry option)

will be ignored for now. In our case we will assume that once the state variable reaches the

upper trigger point it is optimal to abandon the project, and such abandonment policy will not

involve any costs. The disinvestment strategy can be stated as follows:

I ← F (r) + DO(r)

where I takes a positive value since when the firm close its operations will not incur any cost

to disinvest. Obviously, there may be situations where firms have to incur an extra cost when

they want to close, such as the cases of a copper mine or a nuclear power station where some

environmental clean up costs have to be supported. In our case, we want to focus our analysis

on the possibility that some fraction of the lump-sum cost I can be recouped if firms decide

to abandon its operations. Therefore, we will define a new variable α that will measure the

degree of reversibility, i.e., α = I/I. α = 0 corresponds to an option in which the decision

taken is irreversible and can be exercised only once. The case 0 < α < 1 corresponds to partial

reversibility. We will consider three cases: α = 0.25, α = 0.50 and α = 0.75. The case where

α = 1 represents perfect reversibility, a situation that gives rise to a flow option in which two

flows can be switched continuously and costlessly (see Shackleton and Wojakowski (2001))29.

The optimal policy to disinvest is determined using one value matching and one smooth

pasting conditions. This yields a system of two non-linear equations in two variables (C3 and

r):

∫ ∞

0

A(t)e−B(t)r dt + C3ear = I (32a)

temporarily suspend operations incurring no costs and restarting at a later point is quite unrealistic, or even

almost impossible, for many projects. For such cases, suspension is equivalent to an outright abandonment.

The situation where temporary suspension is still possible is analyzed by McDonald and Siegel (1985). In this

case, the owner of a physical capital has a Merton (1973a) European call option to produce a given commodity

at time t, but he can avoid a loss by shutting down permanently the plant if the variable cost of production

exceeds the operating revenues at the option’s maturity.29This limiting situation corresponds to the case where the two threshold will collapse to one common switch-

ing level that will determine the optimal exercise strategy. The strategy change will occur when the so-called

Jorgenson (1963) user costs of capital are equals.

26

∫ ∞

0

−B(t)A(t)e−B(t)r dt + aC3ear = 0 (32b)

Once again, we can compute the values of C3 and r solving the system of the two non-linear

equations numerically, or, as an alternative, we can compute the upper threshold numerically

using the following equation:

I = F (r)− F ′(r)a

(33)

For an active firm, no action will be taken until r reaches the upper barrier r.

6.4 Switching options

The most interesting problem is the one where optimal investment and disinvestment decisions

are considered together. Thus, entry and exit decisions are valued simultaneously originating

a lower bound (r) and an upper bound (r) with r < r, and where an idle firm is induced to

invest once the state variable r crosses the action trigger point r and an active firm will be

induced to disinvest if the state variable crosses the threshold point r. The middle band of

interest rates without entry or exit actions yields what is usually called by economic hysteresis,

since the optimal policy is to maintain the actual status quo, whether the firm is operating or

not. Examples of models of investment and disinvestment policies include the optimal policy

of a mine producing a single homogeneous commodity whose spot price is assumed to follow

an exogenously given geometric Brownian motion (Brennan and Schwartz (1985)), the optimal

entry and exit decisions of a firm using a particular production technology (Dixit (1989a)) and

the problem in a competitive industry with home and foreign firms with entry and exit costs

and where the real exchange rate follows a geometric Brownian motion (Dixit (1989b)).

The corresponding strategy for the entry and exit case can be stated as follows:

I + IO(r) → F (r) + DO(r)

I + IO(r) ← F (r) + DO(r)

In this case, the optimal policy is determined using two value matching and two smooth pasting

conditions resulting in a two-sided (r, r) policy. Other examples of two-sided policies include a

27

dynamic menu cost model with state-dependent pricing in which monetary shocks have system-

atic effects on output (Caplin and Leahy (1991)), a two-asset intertemporal portfolio selection

model incorporating proportional transaction costs and where the investor consumes a fixed

proportion of his wealth in each period of time (Constantinides (1986)), an investment portfolio

choice problem under transaction costs when trading a single risky asset and where the investor

accumulates wealth until some terminal point in time in which he consumes all (Dumas and

Luciano (1991)), a consumer’s decision problem of selling an existing illiquid durable good, such

as a house or a car, to purchase a new one when his wealth fluctuates (Grossman and Laroque

(1990)), an optimal intertemporal consumption and investment policy of an investor facing

both fixed and proportional transaction costs when trading multiple risky assets (Liu (2004))

and a real options holder strategy of continually choosing between two flows (Shackleton and

Wojakowski (2001)).

It is important to note that the investment and disinvestment opportunities at the lower

threshold are, respectively, IO(r = r) = C2ebr and DO(r = r) = C3e

br. Similarly, the in-

vestment and disinvestment opportunities at the upper threshold are, respectively, IO(r =

r) = C2ebr and DO(r = r) = C3e

br. This yields a system of four non-linear equations in four

variables (C2, C3, r and r)30:

I + C2ebr =

∫ ∞

0

A(t)e−B(t)r dt + C3ear (36a)

30It should be noted that this system of four non-linear equations can also be written in the form that is

usually presented in some entry and exit problems (see, for example, Dixit and Pindyck (1994, pg. 218)):

F1(r)− F0(r) = I

F′1(r)− F

′0(r) = 0

F1(r)− F0(r) = I

F′1(r)− F

′0(r) = 0

In our case we are considering that a part of the investment cost I is not sunk, since it can be recouped if the

firm decides to abandon its operations. As a result we have a positive signal in I. But if it was necessary to

pay a lump-sum cost to abandon operations we would have −I. To rule out a money machine of entry and exit

cycles it is necessary that I > I. Thus, the case where α = 1 is just the limiting case.

28

bC2ebr =

∫ ∞

0

−B(t)A(t)e−B(t)r dt + aC3ear (36b)

∫ ∞

0

A(t)e−B(t)r dt + C3ear = I + C2e

br (36c)

∫ ∞

0

−B(t)A(t)e−B(t)r dt + aC3ear = bC2e

br (36d)

The above equations are highly non-linear and as before a closed form solution is not avail-

able. Although we have to rely on numerical methods to get the solution for the two thresholds

and the two constants, such numerical solutions are quite easy to obtain using the numeri-

cal routines for solving simultaneous non-linear equations that are available in many scientific

computing software, such as Mathematica. However, some important economic properties of

the solution can also be obtained by analytical methods similar to the ones employed by Dixit

(1989a). We will use such technique to get a better understanding of the hysteresis effect.

6.5 Economic hysteresis effect

Decisions made under an uncertainty environment where it is costly to reverse economic actions,

will lead to an intermediate range of the state variable, called a hysteresis band, where inaction is

the optimal policy. It turns out that this range of inertia will produce hysteresis, i.e., permanent

effects of temporary shifts. This means that if the state variable starts from a value inside the

hysteretic band (in our case, between r and r), and afterwards crosses one of the action trigger

points and then returns to its previous level, the action will be taken but not reversed. Thus,

hysteresis is an effect that persists after the cause that brought it about has been removed,

i.e., a temporary change will originate a permanent effect. The key issue is that the economic

actions should be taken only when the underlying state variable becomes extremely favorable

and reversed only when its value becomes extremely unfavorable.

Several models of entry and exit decisions of irreversible investments have shown that the

range of inaction can be remarkable large. Such examples of economic hysteresis have been

provided by, among others, Brennan and Schwartz (1985) who have analyzed a firm’s joint

decision of entry and exit a mine that will produce a single homogeneous commodity whose

spot price is assumed to follow a geometric Brownian motion, Dixit (1989a) for the case of a

similar decision of a particular firm using a given production technology and whose output price

29

also follows a geometric Brownian motion and Dixit (1989b) for the case where foreign firms

that have entered in the U.S. market when the dollar was high do not have abandoned their

investments when the dollar started to fall. In this case the state variable is the exchange rate,

namely the price of a dollar in yen, which is assumed to follow once again a geometric Brownian

motion. The economic hysteresis effect is also found to be wide in the optimal consumption

and portfolio choice literature, as it is shown in the intertemporal portfolio selection model

of Constantinides (1986). Therefore, such effect seems to be extremely relevant for many

economic applications. Since interest rates are also an important determinant of investment

and disinvestment decisions it is important to analyze the economic hysteresis effect provoked

by interest rate uncertainty. To our knowledge, this effect has not been previously analyzed

under stochastic interest rates.

The economic hysteresis effect produces a range of values for the state variable that is

usually defined by highly non-linear equations that need numerical solutions. In some cases,

it is possible to use analytic approximations that allow the use of explicit solutions to help

understanding the importance of the hysteresis effect (see, for example, Dixit (1991a)). In

our case, we have highly non-linear equations that use functions with integrals. As a result,

such analytic approximations are very difficult to obtain and we do not attempt to use them.

However, we can established a general property of the solution that yields economic hysteresis.

A procedure like this was previously used by Dixit (1989a) to examined the nature of hysteresis

when the source of uncertainty arises from the output market price, whereas in our case the

uncertainty comes from the stochastic nature of the interest rate term structure. To do so, let

us define the following function:

V (r) = F1(r)− F0(r) (37)

Using the solutions stated by equations (28) and (29) we have:

V (r) = C3ear − C2e

br + F (r) (38)

where F (r) represents the perpetuity value. For small values of r the term with the negative

root b dominates. The term is negative, increasing and concave. For very high interest rate

levels the dominant term is the one associated with the positive root a. The term is positive,

30

increasing and convex. For the intermediate range, it is the perpetuity value that plays a critical

role.

Now, the two value matching and the two smooth pasting conditions can be defined in terms

of V as:

V (r) = I , V ′(r) = 0 , V (r) = I , V ′(r) = 0 (39)

There exits an optimal policy of the type pictured in Figure 13, which is similar in spirit to

the diagrams presented by Constantinides and Richard (1978), Harrison et al. (1983) and Dixit

(1989a, 1991b). The graph seems to indicate that there always exists an unique optimal policy

for the entry and exit decisions problem under stochastic interest rates and this policy is of

a simple form. The existence and form of the optimal policy is addressed using the optimal

control technique of impulse control. From the figure we can also see that the optimum policy

is characterized by the interest rate values r and r. Thus, the firm will invest when it falls to r,

disinvest when it rises to r and continue the status quo over the range (r, r), i.e., an idle firm

does not invest and an active firm does not abandon. To solve the problem we need to adjust

the constants C2 and C3 until the value function V (r) becomes tangent to the horizontal lines

I and I. Then, it turns out that the tangency points are the thresholds r and r.

To get some analytical results it is important to note that:

V ′′(r) < 0 , V ′′(r) > 0 (40)

since V (r) is concave at r and convex at r. Subtracting equation (20) from equation (21) we

see that the function V (r) satisfies the following ordinary differential equation:

1

2σ2∂2V (r)

∂r2− λ

∂V (r)

∂r− V (r) = −1

r(41)

Now evaluating this differential equation at r and using the conditions (39) and (40) we get:

−1

r=

1

2σ2∂2V (r)

∂r2− λ

∂V (r)

∂r− V (r) < −I (42)

Using the same approach at the r we obtain:

−1

r=

1

2σ2∂2V (r)

∂r2 − λ∂V (r)

∂r− V (r) > −I (43)

31

Rearranging we get, respectively:

r < 1/I ≡ Mr (44)

and

r > 1/I ≡ Mr (45)

where Mr and Mr can be viewed, respectively, as the Marshallian trigger interest rates for

investment and disinvestment. Taking decisions using this traditional concept can lead to my-

opic actions since we are implicitly assuming a static expectation for the interest rate dynamics.

Thus, the differences between our thresholds and the Marshallian ones comes from the uncer-

tainty effect. Conditions (44) and (45) highlights that uncertainty widens the Marshallian range

of inaction. In the next section, we will provide some numerical results about this issue. For

now, we will provide some theoretical insights about it.

The economic theory of investment based on the Marshall’s foundations tells us that firms

are induced to invest if the output price exceeds long run average cost and are induced to

suspend or even abandon its operations if the price falls below long run average costs. Let us

suppose that a project can be launched by a firm incurring a sunk cost K which is represented

in our case by I. Once the project is accepted it will return a net operating revenue of one

unit over time, i.e., a perpetuity value. Let this perpetuity be discounted at a rate r, that

represents the opportunity cost of capital in a risk-neutral world. Therefore, the Marshallian

criteria would be to invest when the project has a positive expected net worth, i.e., 1/r > K. It

should be noted that the value of rK, or in our notation rI, represents the full cost of making

and operating the investment, since we are assuming that once the investment is made the

return from the project is identical to a perpetuity making a continuous payment of one unit

over time. The interest rate level that would make one indifferent between investing or not

investing is given by 1/K, or, in our notation, 1/I. As a result, using the Marshall’s economic

foundations we would invest when r was lower than the Marshallian investment trigger point

Mr = 1/I. Similarly, the Marshallian disinvestment trigger point would be Mr = 1/I. Now,

using these two trigger points it is extremely easy to explain the hysteresis nature through a

simple numerical example. Let us suppose that the investment cost is 10 and if a firm wants

32

to abandon operations can recoup 50% of the investment cost (i.e., α = 0.50). As a result,

the investment and disinvestment Marshallian trigger points are, respectively, 0.10 (1/I) and

0.20 (1/I). If the interest rate was initially in the range (0.10, 0.20) and then it falls below the

lower threshold the firm would invest, assuming it was in the idle state. But if the interest rate

returned again to the middle range, this was insufficient to induce the firm to abandon. For

such action, it was necessary that the interest rate would rise above 0.20. Thus, a temporary

change in the interest rate levels has originated a permanent effect. However, the Marshallian

criteria ignores the option value embedded in investment and disinvestment decisions, since the

choice is made between acting right now or do nothing at all, i.e., a now or never decision type.

As a result, the gap between lower and upper thresholds will be even higher, which enhances

the hysteresis effect.

Using these theoretical insights we can consider some limiting cases. We have mentioned

before that we are considering that once the investment commitment has been made, the

investment project return is identical to a perpetuity making a continuous payment of one unit

over time. Thus, no additional resources or expenditures apart from the initial investment

are required to maintain the rights over the project or to sustain the project after it has been

accepted. In addition, if the firm wants to abandon the operating project it can recoup a

fraction α of the investment cost. This means that in our simplifying setting the operating

profits from the project never become negative (the variable costs are zero). As a result, rI

represents the full cost of investment and rαI represents the full disinvestment proceeds. Thus,

our theoretical results have to be restricted to these two issues.

The limiting case where both I and I tend to zero is not very interesting, since both r and

r would diverge. When they tend to extremely large values, both r and r tend to the common

limit 0. But these two issues indicate that sunk costs and disinvestment proceeds are essential

for the hysteresis effect. If we fix α at a level of 0.50 for example, and we impose a higher

investment cost both the lower and the upper thresholds diminish (dr < 0 when dI > 0 and

dr < 0 when dI > 0, respectively). But the upper threshold falls at a higher rate. Thus, there

is a tendency for narrowing the hysteresis band in this case. Let us now fix the investment

cost and analyze the impact of different levels of investment recoup. Maintaining the level of

investment cost fixed and rising the α parameter, gives a very small rise in the lower threshold

33

(dr > 0 when dI > 0) and a sharply decrease in the upper trigger point (dr < 0 when dI > 0).

Thus, the band of inaction will be narrower.

If we set α = 0 the upper threshold diverges to ∞. Thus, an active firm would never

abandon the project. But there is still a finite interest rate level that will induce an idle firm

to invest in a project that afterwards will not be abandoned. This issue derives from the fact

that the profits from the project never become negative. As a result, the firm will continue

its operations. It should be noted that in this case (α = 0) the constant C3 goes to zero in

equations (36a) - (36d), since the option to disinvest becomes worthless. Thus, equations (36a)

and (36b) turn out to be the equations (30a) and (30b). As a result, the lower threshold for the

case where α = 0 can be computed using equation (31). Once the state variable r crosses this

lower trigger point an idle firm will invest immediately and will maintain its operations forever.

An active firm will never abandon. But if there were some variable costs these could lead to

negative profits and then there would be an upper trigger point that would induce the firm to

shut down optimally. Similarly, if the investment costs to reenter again (if a firm decides to

shut down its operations previously) goes to ∞, the option to reinvest becomes worthless and,

as a result, the constant C2 goes to zero in equations (36a) - (36d). In this case, equations

(36c) and (36d) turn out to be the equations (32a) and (32b). Therefore, the upper threshold

point is computed using equation (33). An idle firm never invests due to high entrance costs

and an operating firm may be induced to optimally abandon its operations when the interest

rate crosses the upper trigger point. This point indicates how bad things must be, i.e., how

high interest rates should rise before an active firm abandon its operations, since it knows that

due to high entrance costs it can never reinvest later again.

If interest rate uncertainty goes to zero, then r → Mr and r → Mr. This imply that

without uncertainty the range of inaction is determined by the Marshallian trigger points. As

the interest rate volatility starts rising, the lower threshold will fall and the upper threshold

will rise, which will lead to a wider hysteresis band. Now, maintaining the volatility fixed at a

positive level and letting I → 0, we have dr/dI → −∞ and dr/dI →∞. This means that when

there is some level of interest rate uncertainty, the hysteresis level emerges very quickly even for

very small investment costs. This also means that apart from the output price uncertainty (see,

for example, Dixit (1989a)), the interest rate uncertainty also plays a critical role for widening

34

the hysteretic band. All these theoretical insights can be confirmed with the numerical results

that we present in the next section.

We know that, on one hand side, in practice firms do not invest until price rises substantially

above long run cost. In fact, it is not difficult to find situations where firms require hurdle rates

that are much higher than the cost of capital or the internal rate of return. On the other

hand side, firms will continue its operations even when they are facing operating losses, and

will only disinvest when the output price falls substantially bellow the variable cost. Thus,

this upside and downside aspect of the decisions taken by firms may be possibly explained by

the economic hysteresis phenomena. In fact, looking at conditions (44) and (45) it is proved

that the Marshallian trigger points are not optimal if waiting for better information about

the interest rate behaviour is possible. This explains why our lower threshold interest rate is

smaller than the Marshallian trigger entry level and why our critical point of abandonment

exceeds the Marshallian trigger exit point. For example, at the lower Marshallian trigger point

the investment opportunity is equivalent to an option that is only just in the money. It is

not optimal for the firm to exercise it unless it goes deeper in the money. Since immediate

action has an opportunity cost, namely loss of the option to wait for better information, the

economic hysteresis effect is enhanced when waiting is possible. It should be noted that most

of the upside potential remains even if an economic action is not taken immediately, but only

after a small time delay. Thus, it is the possibility of a downturn and the ability to avoid

a wrong economic action that makes waiting valuable. In other words, it is the downside

uncertainty that matters most for optimal investment decisions when waiting is possible. This

is an implication of the ”bad news principle” of Bernanke (1983a). By waiting, the firm can

improve its chances of making the correct decision. Thus, under ongoing uncertainty and when

waiting is possible the zone of inaction that takes option values into account is wider. We may

conclude that when interest rate uncertainty is a key factor for capital budgeting decisions, idle

firms are more reluctant to invest and active firms are more reluctant to exit. Thus, we may

claim that the reversibility degree and the interest rate uncertainty can help explain investment

and disinvestment cyclical fluctuations. Moreover, the analysis of the effect of option values

under interest rate uncertainty on investment and disinvestment decisions can also help explain

why there is a gap in practice between required hurdle rates and the cost of capital.

35

6.6 Numerical analysis

After providing some theoretical insights about the economic hysteresis effect under stochastic

interest rates, let us now proceed with some numerical results that will confirm our analytical

results. We will consider the base case volatility level, σ = 0.0854, and a smaller and a higher

volatility (σ = 0.03 and σ = 0.3, respectively) for comparative purposes. In addition, we use

two different levels of investment cost, I = 10 and I = 7.5. We also establish three degrees of

reversibility, α = 0.25, α = 0.50 and α = 0.75. We also present the case of perfect reversibility,

α = 1.00, to illustrate the limiting case. The case where α = 0 is also illustrated since it falls

in the single investment strategy situation.

It should be noted that, for now, we are considering the no mean-reverting case. Therefore,

we have to use equations (14) and (15) to compute the perpetuity and the derivative of the

perpetuity functions, respectively. In addition, we have to set a fixed upper limit for the

integrals in both functions, otherwise their values would be +∞ and −∞, respectively. A

question now arises. What value T should we use at the upper limits of the integrals? We

have tried several time values such as T = 100, T = 500, T = 1000, T = 3000, etc. It turns

out that from a practical point of view any of these values can be considered as a sufficient

large maturity, which gives a time-independent resemblance for the problem. For the single

investment strategy the choice of T is not sensible for the lower threshold point. For the

disinvestment single strategy, however, this is not the case. Indeed, the upper threshold is rising

with T , although not to much strongly marked, especially for low and moderate volatility levels.

But for the combined entry and exit strategy, which is the most interesting one, the use of any

of those T values does not produce any significative change on both lower and upper trigger

points. Yet, we present two different time values, T = 500 and T = 1000, which confirms what

we have described.

Table 4 presents the lower trigger points for the single investment strategy considering

different investment cost levels and different interest rate volatilities. Under this strategy,

an active firm never shuts its project, since the profits never become negative. Thus, these

thresholds indicate which is the interest rate level that will induce an idle firm to enter in a

project and continue its operations forever since the option to shut down is worthless (i.e., it

corresponds to the case where α = 0). For example, considering the base case volatility level

36

and I = 10 it would be necessary that the interest rate falls to 1.94% to induce an idle firm

to invest. At this rate level, the firm will change the full cost of investment (i.e., option to

invest plus investment cost) by a project paying a perpetuity. From the table we can conclude

that: (i) for a given level of volatility the lower threshold falls as the investment cost rises;

and (ii) for a given investment cost level the lower trigger point falls as the volatility level

rises. In addition, all investment thresholds when uncertainty is considered are lower than the

Marshallian investment trigger point (0.10 and 0.1333 for I = 10 and I = 7.5, respectively). It

should be noted that for high levels of uncertainty firms would never invest.

Table 4: Lower thresholds for the investment option of

the special case under different levels of investment costs

and interest rate volatility.

I = 10 I = 7.5

T σ = 0.03 σ = 0.0854 σ = 0.3 σ = 0.03 σ = 0.0854 σ = 0.3

500 0.0809 0.0194 −0.4470 0.1145 0.0642 −0.3450

1000 0.0809 0.0194 −0.4470 0.1145 0.0642 −0.3450

Figure 14 presents the project’s value rights (i.e., the value of the option to invest) and the

NPV from the project for the volatility level σ = 0.0854. For greater visual appeal we present

the functions for T = 100. It is easily seen that for high interest rate values firms should not

invest since the value of rights are greater than the NPV of the project. The interest rate level

that turns the traditional NPV equal to zero is r = 0.1833. But at this rate level the rights from

the project have a value of 4.3384. Even if interest rates fall bellow r = 0.1833, which would

turn the NPV positive, it is not optimal that an idle firm invests immediately, because waiting

for better information about the term structure dynamics has value. Therefore, it would be

necessary that the interest rate level falls to the optimal trigger point r = 0.0194 to induce an

idle firm to invest.

Table 5 presents the upper trigger points for the single disinvestment strategy considering

different investment cost levels and interest rate volatilities and different fractions of disin-

vestment proceeds to investment costs. This case corresponds to the one where the option

37

to reinvest becomes worthless since the investment costs to reenter again are extremely high.

Thus, idle firms do not invest and active firms will only disinvest if interest rates goes to suf-

ficiently high values, because they know they cannot reinvest again. Considering the volatility

level of σ = 0.0854, an investment cost of I = 10 and α = 0.50, it would be necessary that

interest rates would be around 0.40 to induce an active firm to abandon its operations, knowing

that its action could not be reversed later. From the table we can take the following conclu-

sions: (i) for a given level of volatility and investment cost the upper threshold falls when the

parameter α rises (i.e., the disinvestment proceeds rises); (ii) for a given volatility level and α

parameter the upper trigger point falls as the investment cost rises (at the end it originates

a rise on the disinvestment proceeds); and (iii) for a given investment cost and α parameter

the optimal disinvestment threshold rises as volatility rises. All disinvestment thresholds when

uncertainty is considered are higher than the Marshallian disinvestment trigger points. Thus,

higher uncertainty produces higher upper trigger points. It should be noted that for high levels

of uncertainty firms would never disinvest almost for sure. Even for moderate and low levels

of uncertainty it would be necessary that firms could recoup a very high fraction of the invest-

ment costs to induce firms to abandon permanently their operations. This issue arises from the

fact that operating profits can never become negative and there is no option to reenter again.

As we will see below, when the option to invest again is considered these thresholds will fall

significantly due to the reentry option effect.

Table 5: Upper thresholds for the disinvestment option

of the special case under different levels of disinvestment

proceeds and interest rate volatility and different ratios

of the disinvestment proceeds to the investment costs.

I = 10 I = 7.5

T α σ = 0.03 σ = 0.0854 σ = 0.3 σ = 0.03 σ = 0.0854 σ = 0.3

500 0.25 0.4225 0.5091 1.3648 0.5555 0.6138 1.4557

1000 0.4225 0.5286 1.4998 0.5555 0.6222 1.5871

500 0.50 0.2251 0.3818 1.1746 0.2900 0.4229 1.2499

1000 0.2260 0.4145 1.3148 0.2901 0.4520 1.3884

38

500 0.75 0.1659 0.3365 1.0739 0.2044 0.3675 1.1447

1000 0.1707 0.3723 1.2160 0.2062 0.4013 1.2856

500 1.00 0.1402 0.3095 1.0053 0.1659 0.3365 1.0739

1000 0.1479 0.3467 1.1484 0.1707 0.3723 1.2160

Figure 15 presents the project’s exit rights (i.e., the value of the option to abandon) and the

NPV of the abandonment decision for the volatility level σ = 0.0854, α = 0.50 and T = 500.

The convex function in the graph represents the abandonment option (i.e., C3ear) and the

concave function pictures the NPV of the abandonment decision (i.e., I − F (r)). We can

see that for low interest rate levels firms should not disinvest since it is better to keep the

option to abandon alive. For low rate values the disinvestment proceeds are lower than the

operating profits from the project which turns the NPV of the decision to abandon negative.

The interest rate level that turns the traditional NPV of the decision to abandon equal to zero

is r = 0.3373. But at this rate level the exit rights from the project have a value of 0.6475.

Thus, the firm prefers to keep its abandonment option alive. Even if interest rates rise a little

above r = 0.3373, which would turn the NPV of the abandonment decision positive, it is not

optimal that an active firm disinvests immediately. Therefore, it would be necessary that the

interest rate level rises to the optimal trigger point r = 0.3818 to induce an active firm to shut

down permanently its operations knowing that the reentry investment cost makes the decision

to reinvest later prohibitive.

Looking to the graph we can note that the convex curve representing the option to abandon

lies above the concave function representing the NPV of the abandonment decision to the right

of the upper trigger point r. Thus, someone could wonder if the disinvestment decision is

optimal only at the upper threshold, and keeping the option to abandon alive the preferred

policy for higher interest rate values. Obviously, this thinking is not correct because the exit

option has only a valid economic interpretation for interest rate levels lower that the upper

trigger point. If this was not the case it would be created a pure speculative bubble. Thus, the

value of keeping the exit option alive would be very high because the possibility of reaching

a higher interest rate would offer an even higher value of the abandonment option, with no

39

disinvestment action ever in sight. This would turn the value of waiting for better information

infinite, which is not economically intuitive. If this was the case, firms would never invest and

disinvest because it was always better to wait for better information. In conclusion, the entry

and exit options only have a valid interpretation above r and below r, respectively.

Tables 6 and 7 presents both the lower and upper trigger points for the entry and exit

combined strategy considering different investment cost levels, I = 10 for table 6 and I = 7.5

for table 7 and different interest rate volatilities and fractions of disinvestment proceeds to

investment costs. This case corresponds to the one where idle firms are induced to invest if

the interest rate value falls to a sufficient low level, but they own an option to abandon later if

interest rates rise to very high values. Once the project is abandoned, firms own an option to

reinvest again if interest rates reverse to very low levels again. It turns out that this combined

strategy originates a range where inaction is the optimal policy, i.e., idle firms do not invest

and active firms do not abandon their operations. Considering a volatility level of σ = 0.0854,

an investment cost of I = 10 and α = 0.50, the lower and upper trigger points are, respectively,

r = 0.0199 and r = 0.2641, which originates a range of inaction of 0.2442 (r − r). Figure 16

depicts this numerical example. It is possible to see that an unique optimal solution exists. In

this case, the Marshallian trigger points would be Mr = 1/I = 0.10 and Mr = 1/I = 0.20,

originating a band of inaction of 0.10 (Mr − Mr). In this case, our lower trigger point is

approximately 80 percent below the Marshallian investment threshold and our upper trigger

point is approximately 32 percent above the corresponding Marshallian exit point. Therefore,

uncertainty and the embedded option values are responsible for the hysteretic band widening.

Considering these parameters, it is possible to see that the lower trigger point for the single

investment strategy (i.e., without the exit option) is r = 0.0194 (see table 4). For the combined

strategy, this lower trigger point is now r = 0.0199 (see table 6). The small difference in value

arises from the firm’s possibility to shut down later if interest rates start rising for very high

levels. Similarly, the upper trigger point for the single disinvestment strategy (i.e., without the

reentry option) is r = 0.3818 (see table 5). For the combined strategy, the upper trigger point

is now r = 0.2641 (see table 6). This difference comes from the value of the reentry option

that is owned by the firm. But now the difference is much more pronounced. There is an

economic explanation for this fact. Thus, when the firm invests in a project and has an option

40

to shut down in the future, the disinvestment proceeds present value is very small because the

prospect of close its operations is sufficiently far in the future. As a result, the impact on the

lower trigger point is very diminutive. However, when a firm is operating and decides to shut

down its activities it will receive almost immediately the disinvestment proceeds value, which

originates a bigger present value and justifies the greater differences between the two upper

trigger points.

We can take the following conclusions from the tables: (i) for a given level of volatility and

investment cost the lower threshold rises and the upper threshold falls when the parameter α

rises (i.e., the disinvestment proceeds rises). Thus, the range of inaction will be narrower; (ii)

for a given volatility level and α parameter both the lower and upper trigger points falls as the

investment cost rises (at the end it originates a rise on the disinvestment proceeds); and (iii)

for a given investment cost and α parameter the optimal lower trigger point falls and the upper

threshold rises as volatility rises, widening the hysteretic range31.

Table 6: Upper and lower thresholds for the switching

option of the special case under an investment cost of 10

for different ratios of the disinvestment proceeds to the

investment costs and different interest rate volatilities.

I = 10

σ = 0.03 σ = 0.0854 σ = 0.3

T α r r r r r r

500 0.25 0.0809 0.4225 0.0194 0.4725 −0.4467 0.5922

1000 0.0809 0.4225 0.0194 0.4725 −0.4467 0.5922

500 0.50 0.0810 0.2239 0.0199 0.2641 −0.4427 0.2217

1000 0.0810 0.2239 0.0199 0.2641 −0.4427 0.2217

31It should be noted that this is not always true for the upper trigger point. Thus, for very high values of

α and investment costs, which produces high disinvestment proceeds, the upper threshold starts rising as the

volatility rises, but for very high volatility levels the upper threshold turns its behaviour and starts falling. A

possible explanation for this issue may come from the ”bird-in-the-hand” argument. Thus, since uncertainty is

so high and the value of the disinvestment proceeds is significant, firms may be induced to shut down at lower

rates in order to get a safe present value of the disinvestment proceeds.

41

500 0.75 0.0815 0.1556 0.0233 0.1717 −0.4273 0.0061

1000 0.0815 0.1556 0.0233 0.1717 −0.4273 0.0061

500 1.00 0.0991 0.0991 0.0671 0.0671 −0.2872 −0.2872

1000 0.0991 0.0991 0.0671 0.0671 −0.2872 −0.2872

Table 7: Upper and lower thresholds for the switching

option of the special case under an investment cost of 7.5

for different ratios of the disinvestment proceeds to the

investment costs and different interest rate volatilities.

I = 7.5

σ = 0.03 σ = 0.0854 σ = 0.3

T α r r r r r r

500 0.25 0.1145 0.5555 0.0642 0.6029 −0.3447 0.7614

1000 0.1145 0.5555 0.0642 0.6029 −0.3447 0.7614

500 0.50 0.1145 0.2899 0.0645 0.3355 −0.3412 0.3497

1000 0.1145 0.2899 0.0645 0.3355 −0.3412 0.3497

500 0.75 0.1148 0.2005 0.0676 0.2271 −0.3262 0.1200

1000 0.1148 0.2005 0.0676 0.2271 −0.3262 0.1200

500 1.00 0.1331 0.1331 0.1127 0.1127 −0.1838 −0.1838

1000 0.1331 0.1331 0.1127 0.1127 −0.1838 −0.1838

We know that it is extremely complicated to analyze analytically the impact of the σ and λ

parameters on the optimal trigger points due to the effects they produce on the value function

V (r), because both enter in the quadratic equation with roots a and b. However, we can resort

some numerical simulations that can highlight their effects. For greater visual appeal we show

the corresponding pictures as continuous curves and not as step functions. Figure 17 presents

the impact on the entry and exit thresholds as the volatility rises, considering I = 10, α = 0.5

42

and λ = 0.0. Clearly, there is a tendency to a wider range of inaction as the volatility rises,

as we have already mentioned before. Figure 18 shows the impact on the upper and lower

trigger points as the price of interest rate risk rises in absolute value, considering I = 10,

α = 0.5 and σ = 0.0854. It should be noted that positive premiums exist if λ < 0, but for

easy visualization we use the parameter absolute value in this simulation. Thus, as the price of

interest rate risk rises there is a tendency for a fall in the upper trigger point and a rise in the

lower threshold, which originates a narrower hysteretic band. There is an economic explanation

for this. Under the equivalent martingale economy, a higher positive term premia indicates that

long-term rates are high relative to the instantaneous rate. This means that interest rates are

expected to increase. Thus, with higher positive premiums it is more important to make the

investment before the likely rise in the interest rate (see Ingersoll and Ross (1992)). This

explains the tendency for a rise in the lower threshold. A similar argument can be used for the

upper trigger point. Thus, with the expected increase in the interest rates firms are induced to

abandon earlier otherwise the present value of the disinvestment proceeds will be lower due to

the prospect of a rise in the interest rate. As a result, the upper threshold has a tendency to

fall.

7 Conclusions

When interest rates fall, firms make durable investments, that is to say that they switch from

cash (an immediate asset) to longer lived assets with cash flows further ahead in time. When

interest rates rise, they will stop undertaking any durable projects. Furthermore if flexibility

exists they will also try and reverse the investment process, i.e. divest away from projects with

long lived cash flows into projects with more immediate payoffs.

We model this investment hysteresis explicitly using the most tractable form of interest rate

uncertainty and describe the sensitivity of the interest rate band to the model parameters. We

do this to analyze the beneficial effects of waiting to invest in the presence of uncertainty as

well as to shed light on the macro influence of interest rate changes on investment policies.

43

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

6

8

10

12

14

16

18

Interest rate (r)

Per

petu

ity (

F(r

))

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

20

25

Interest rate (r)

Per

petu

ity (

F(r

))

σ0

σ1

σ2

Figure 1: Value of a perpetuity as a function of

the interest rate using the base case parameter

values.

Figure 2: Value of a perpetuity as a function

of the interest rate using the base case parame-

ter values for different levels of volatility (σ0 =

0.0854, σ1 = 0.03 and σ2 = 0.3).

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.58

10

12

14

16

18

20

22

24

26

Interest rate volatility (σ)

Per

petu

ity (

F(r

))

λ0

λ1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

50

100

150

200

250

300

350

400

450

Interest rate volatility (σ)

Per

petu

ity (

F(r

))

λ0

λ1

Figure 3: Value of a perpetuity as a function of

the interest rate volatility using the base case pa-

rameter values for different levels of risk premium

(λ0 = 0 and λ1 = −0.1) and r(0) = 0.06.

Figure 4: Value of a perpetuity as a function of

the interest rate volatility using the special case

parameter values for different levels of risk pre-

mium (λ0 = 0 and λ1 = −0.1) and r(0) = 0.06.

44

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

6

8

10

12

14

16

18

20

Interest rate (r)

Per

petu

ity (

F(r

))

σ0

σ1

σ2

σ3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

6

8

10

12

14

16

18

20

Interest rate (r)

Per

petu

ity (

F(r

))

σ0 σ

1 σ

2

σ3

Figure 5: Value of a perpetuity as a function

of the interest rate using the base case parame-

ter values for different levels of volatility (σ0 =

0.0854, σ1 = 0.03, σ2 = 0.3 and σ3 = 0).

Figure 6: Value of a perpetuity as a function

of the interest rate using the special case pa-

rameter values for different levels of volatility

(σ0 = 0.0854, σ1 = 0.03, σ2 = 0.3 and σ3 = 0).

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

6

8

10

12

14

16

18

Interest rate (r)

Per

petu

ity (

F(r

))

λ0

λ1

λ2

−0.5 −0.45 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 00

10

20

30

40

50

60

70

80

90

100

Price of interest rate risk (λ)

Per

petu

ity (

F(r

))

κ = 0.2339

κ = 0

Figure 7: Value of a perpetuity as a function of

the interest rate using the base case parameter

values for different levels of risk premium (λ0 =

0.0, λ1 = −0.1 and λ2 = −0.2).

Figure 8: Value of a perpetuity as a function of

the risk premium using the base and special cases

parameter values and r(0) = 0.06.

45

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

20

25

Interest rate (r)

Per

petu

ity (

F(r

))

κ0 κ

1 κ

2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

5

10

15

20

25

30

Speed of adjustment (κ)

Per

petu

ity (

F(r

)) r(0)1

r(0)2

r(0)3

Figure 9: Value of a perpetuity as a function of

the interest rate using the base case parameter

values for different levels of speed of adjustment

(κ0 = 0.2339, κ1 = 0.4 and κ2 = 0.1).

Figure 10: Value of a perpetuity as a function

of the speed of adjustment using the base case

parameter values for different levels of interest

rate (r(0)1 = 0.02, r(0)2 = 0.1241 and r(0)3 =

0.25).

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

10

20

30

40

50

60

Interest rate (r)

Per

petu

ity (

F(r

))

θ0

θ1

θ2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

6

8

10

12

14

16

18

20

Interest rate (r)

Per

petu

ity (

F(r

))

κ = 0.2339

κ = 0

Figure 11: Value of a perpetuity as a function

of the interest rate using the base case parameter

values for different levels of asymptotic interest

rate (θ0 = 0.0808, θ1 = 0.15 and θ2 = 0.02).

Figure 12: Value of a perpetuity as a function of

the interest rate using the base and special cases

parameter values.

46

r

V(r)

r r

I

I

Figure 13: Determination of the theoretical upper and lower thresholds.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−20

0

20

40

60

80

100

Interest rate (r)

C2 E

xp(b

r), F

(r)

− I

0 0.1 0.2 0.3 0.4 0.5 0.6−20

−15

−10

−5

0

5

10

15

20

Interest rate (r)

C3 E

xp(a

r), α

× I

− F

(r)

Figure 14: Value of entry rights and NPV of the

project.

Figure 15: Value of exit rights and NPV of the

abandonment decision.

47

−0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

2

4

6

8

10

12

14

Interest rate (r)

V(r

)

Figure 16: Determination of the numerical upper and lower thresholds.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Interest rate volatility (σ)

Upp

er a

nd lo

wer

thre

shol

ds

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.05

0.1

0.15

0.2

0.25

0.3

Price of interest rate risk (λ)

Upp

er a

nd lo

wer

thre

shol

ds

Figure 17: Entry and exit thresholds as func-

tions of interest rate volatility.

Figure 18: Entry and exit thresholds as func-

tions of interest rate risk price.

48

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